Partition polynomials in three of Jian Zhou’s papers

The papers are A) “On Topological 1D Gravity. I” and B) “On Emergent Geometry of the Gromov-Witten Theory of Quintic Calabi-Yau Threefold” and C) “Topological 1D Gravity, KP Hierarchy, and Orbifold Euler Characteristics of ” coauthored with Zhiyuan Wang.

In A:

Page 9 contains the partition polynomials (PrPs) of OEIS A145271.

The first equality in Sec. 2.12 on p. 17 contain the PrPs of A248927.

Unsigned coefficients and monomials of the PrPs of A134685 are in the numerators on p. 19 for the derivatives of eqn. 60.

The PrPs above are avatars of Lagrange compositional inversion.

Apparently, the PrPs of A036038 are on p. 21 with an interpretation as sums over rooted trees.

A036040 on p. 69.

The derivatives in eqn. 257 on p. 73 are similar if not identical to those of the elementary Schur polynomials of A130561 and the related compositional partition polynomials that they easily scale to such as the set of PrPs just above..

(Eqn. 262 on p. 74 is an integrable hierarchy related to the inviscid Hopf-Burgers equation, which I note in a recent post here.)

(Eqn. 269 on p. 75 has the generalized creation op related to another recent post here.)

In B:

A134264 is on pp. 27 and 29.

A035206 is on p. 29, which can be generated by taking the derivative of the PrPs of A134264 w.r.t. the indeterminate .

(The section “5.4 Deformation of flow equations” on p. 57 and Theorem 5.4 on pp. 66 and 67 have differential equations related to the integrable hierarchy of the inviscid Hopf-Burgers equation, which I discuss in a recent post.) 

In C:

A036040 (every other row) on p. 5.

A036039 on pp. 26 and 27.